Question

determine if the given pair of functions are inverse functions of each other using the composition cancellation equations.

$f(x)=(7 + x)^{2},x\geq - 7$ and $g(x)=\sqrt{x}+7$

select the correct choice and fill in the answer boxes within your choice.

(simplify your answers)

$\bigcirc$ a. $f(x)$ and $g(x)$ are not inverse functions of each other because $f(g(x))=\square$ and $g(f(x))=\square$

$\bigcirc$ b. $f(x)$ and $g(x)$ are inverse functions of each other because $f(g(x))=\square$ and $g(f(x))=\square$

Explanation:

Step1: Calculate \(f(g(x))\)

Substitute \(g(x)=\sqrt{x}+7\) into \(f(x)\).

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Step2: Calculate \(g(f(x))\)

Substitute \(f(x)=(7 + x)^{2}\) into \(g(x)\).

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Since \(x\geq - 7\), then \(\vert7 + x\vert=x + 7\). So \(g(f(x))=(x + 7)+7=x+14\)

Answer:

A. \(f(x)\) and \(g(x)\) are not inverse functions of each other because \(f(g(x))=x + 28\sqrt{x}+196\) and \(g(f(x))=x + 14\)